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Chapter 35 NAME
Externalities
Introduction. When there are externalities, the outcome from indepen-
dently chosen actions is typically not Pareto efficient. In these exercises,
you explore the consequences of alternative mechanisms and institutional
arrangements for dealing with externalities.
Example: A large factory pumps its waste into a nearby lake. The lake
is also used for recreation by 1,000 people. Let Xbe the amount of waste
that the firm pumps into the lake. Let Yibe the number of hours per day
that person ispends swimming and boating in the lake, and let Cibe the
number of dollars that person ispends on consumption goods. If the firm
pumps Xunits of waste into the lake, its profits will be 1,200X−100X2.
Consumers have identical utility functions, U(Yi,C
i,X)=Ci+9Yi−
Y2
i−XYi, and identical incomes. Suppose that there are no restrictions
on pumping waste into the lake and there is no charge to consumers for
using the lake. Also, suppose that the factory and the consumers make
their decisions independently. The factory will maximize its profits by
choosing X= 6. (Set the derivative of profits with respect to Xequal
to zero.) When X= 6, each consumer maximizes utility by choosing
Yi=1.5. (Set the derivative of utility with respect to Yiequal to zero.)
Notice from the utility functions that when each person is spending 1.5
hours a day in the lake, she will be willing to pay 1.5 dollars to reduce
Xby 1 unit. Since there are 1,000 people, the total amount that people
will be willing to pay to reduce the amount of waste by 1 unit is $1,500.
If the amount of waste is reduced from 6 to 5 units, the factory’s profits
will fall from $3,600 to $3,500. Evidently the consumers could afford to
bribe the factory to reduce its waste production by 1 unit.
35.1 (2) The picturesque village of Horsehead, Massachusetts, lies on a
bay that is inhabited by the delectable crustacean, homarus americanus,
also known as the lobster. The town council of Horsehead issues permits
to trap lobsters and is trying to determine how many permits to issue.
The economics of the situation is this:
1. It costs $2,000 dollars a month to operate a lobster boat.
2. If there are xboats operating in Horsehead Bay, the total revenue
from the lobster catch per month will be f(x)=$1,000(10x−x2).
(a) In the graph below, plot the curves for the average product, AP (x)=
428 EXTERNALITIES (Ch. 35)
2 4 6 8 10 12
2
4
6
8
10
12
x
AP, MP
0
AP
Cost
MP
(b) If the permits are free of charge, how many boats will trap lobsters
in Horsehead, Massachusetts? (Hint: How many boats must enter before
(c) What number of boats maximizes total profits? Set MP
(d) If Horsehead, Massachusetts, wants to restrict the number of boats to
the number that maximizes total profits, how much should it charge per
month for a lobstering permit? (Hint: With a license fee of Fthousand
dollars per month, the marginal cost of operating a boat for a month
35.2 (2) Suppose that a honey farm is located next to an apple orchard
and each acts as a competitive firm. Let the amount of apples produced
be measured by Aand the amount of honey produced be measured by H.
The cost functions of the two firms are cH(H)=H2/100 and cA(A)=
A2/100 −H. The price of honey is $2 and the price of apples is $3.
(a) If the firms each operate independently, the equilibrium amount of
NAME 429
(b) Suppose that the honey and apple firms merged. What would be
(c) What is the socially efficient output of honey? 150. If the firms
stayed separate, how much would honey production have to be subsidized
35.3 (2) In El Carburetor, California, population 1,001, there is not
much to do except to drive your car around town. Everybody in town
is just like everybody else. While everybody likes to drive, everybody
complains about the congestion, noise, and pollution caused by traffic. A
typical resident’s utility function is U(m, d, h)=m+16d−d2−6h/1,000,
where mis the resident’s daily consumption of Big Macs, dis the number
of hours per day that he, himself, drives, and his the total amount of
driving (measured in person-hours per day) done by all other residents
of El Carburetor. The price of Big Macs is $1 each. Every person in El
Carburetor has an income of $40 per day. To keep calculations simple,
suppose it costs nothing to drive a car.
(a) If an individual believes that the amount of driving he does won’t af-
fect the amount that others drive, how many hours per day will he choose
to drive? 8. (Hint: What value of dmaximizes U(m, d, h)?)
(b) If everybody chooses his best d, then what is the total amount hof
(d) If everybody drives 6 hours a day, what will be the utility level of a
(e) Suppose that the residents decided to pass a law restricting the total
number of hours that anyone is allowed to drive. How much driving
should everyone be allowed if the objective is to maximize the utility of
the typical resident? (Hint: Rewrite the utility function, substituting
430 EXTERNALITIES (Ch. 35)
(f) The same objective could be achieved with a tax on driving. How
much would the tax have to be per hour of driving? (Hint: This price
would have to equal an individual’s marginal rate of substitution between
35.4 (3) Tom and Jerry are roommates. They spend a total of 80 hours
a week together in their room. Tom likes loud music, even when he sleeps.
His utility function is UT(CT,M)=CT+M,whereCTis the number
of cookies he eats per week and Mis the number of hours of loud music
per week that is played while he is in their room. Jerry hates all kinds
of music. His utility function is U(CJ,M)=CJ−M2/12. Every week,
Tom and Jerry each get two dozen chocolate chip cookies sent from home.
They have no other source of cookies. We can describe this situation with
a box that looks like an Edgeworth box. The box has cookies on the
horizontal axis and hours of music on the vertical axis. Since cookies are
private goods, the number of cookies that Tom consumes per week plus
the number that Jerry consumes per week must equal 48. But music in
their room is a public good. Each must consume the same number of
hours of music, whether he likes it or not. In the box, let the height of a
point represent the total number of hours of music played in their room
per week. Let the distance of the point from the left side of the box be
“cookies for Tom” and the distance of the point from the right side of the
box be “cookies for Jerry.”
0122436
48
20
40
60
Cookies
Music
80
Blue Line
Red Line
Blue Shading
a
b
Blue Line
Red Line
Tom
Jerry
(a) Suppose the dorm’s policy is that you must have your roommate’s
permission to play music. The initial endowment in this case denotes the
situation if Tom and Jerry make no deals. There would be no music, and
each person would consume 2 dozen cookies a week. Mark this initial
endowment on the box above with the label A. Use red ink to sketch
the indifference curve for Tom that passes through this point, and use
NAME 431
blue ink to sketch the indifference curve for Jerry that passes through
this point. [Hint: When you draw Jerry’s indifference curve, remember
two things: (1) He hates music, so he prefers lower points on the graph
(b) Suppose, alternatively, that the dorm’s policy is “rock-n-roll is good
for the soul.” You don’t need your roommate’s permission to play music.
Then the initial endowment is one in which Tom plays music for all of
the 80 hours per week that they are in the room together and where each
consumes 2 dozen cookies per week. Mark this endowment point in the
Calculus 35.5 (0) A clothing store and a jewelry store are located side by side
in a small shopping mall. The number of customers who come to the
shopping mall intending to shop at either store depends on the amount
of money that the store spends on advertising per day. Each store also
attracts some customers who came to shop at the neighboring store. If
the clothing store spends $xCper day on advertising, and the jeweller
spends $xJon advertising per day, then the total profits per day of the
clothing store are ΠC(xC,x
J)=(60+xJ)xC−2x2
C, and the total profits
per day of the jewelry store are ΠJ(xC,x
J) = (105 + xC)xJ−2x2
J.(In
each case, these are profits net of all costs, including advertising.)
(a) If each store believes that the other store’s amount of advertising
is independent of its own advertising expenditure, then we can find the
equilibrium amount of advertising for each store by solving two equations
in two unknowns. One of these equations says that the derivative of the
clothing store’s profits with respect to its own advertising is zero. The
other equation requires that the derivative of the jeweller’s profits with
respect to its own advertising is zero. These two equations are written as
432 EXTERNALITIES (Ch. 35)
(b) The extra profit that the jeweller would get from an extra dollar’s
worth of advertising by the clothing store is approximately equal to the
derivative of the jeweller’s profits with respect to the clothing store’s ad-
vertising expenditure. When the two stores are doing the equilibrium
amount of advertising that you calculated above, a dollar’s worth of ad-
vertising by the clothing store would give the jeweller an extra profit of
(c) Suppose that the owner of the clothing store knows the profit functions
of both stores. She reasons to herself as follows. Suppose that I can
decide how much advertising I will do before the jeweller decides what
he is going to do. When I tell him what I am doing, he will have to
adjust his behavior accordingly. I can calculate his reaction function to
my choice of xC, by setting the derivative of his profits with respect to his
own advertising equal to zero and solving for his amount of advertising
as a function of my own advertising. When I do this, I find that xJ=
(d) Suppose that the clothing store and the jewelry store have the same
profit functions as before but are owned by a single firm that chooses
the amounts of advertising so as to maximize the sum of the two stores’
profits. The single firm would choose xC=$37.50 and xJ=
$45. Without calculating actual profits, can you determine whether
total profits will be higher, lower, or the same as total profits would be
35.6 (2) The cottagers on the shores of Lake Invidious are an unsavory
bunch. There are 100 of them, and they live in a circle around the lake.
Each cottager has two neighbors, one on his right and one on his left.
There is only one commodity, and they all consume it on their front
lawns in full view of their two neighbors. Each cottager likes to consume
the commodity but is very envious of consumption by the neighbor on
his left. Curiously, nobody cares what the neighbor on his right is doing.
In fact every consumer has a utility function U(c, l)=c−l2,wherecis
his own consumption and lis consumption by his neighbor on the left.
428 EXTERNALITIES (Ch. 35)
2 4 6 8 10 12
2
4
6
8
10
12
x
AP, MP
0
AP
Cost
MP
(b) If the permits are free of charge, how many boats will trap lobsters
in Horsehead, Massachusetts? (Hint: How many boats must enter before
(c) What number of boats maximizes total profits? Set MP
(d) If Horsehead, Massachusetts, wants to restrict the number of boats to
the number that maximizes total profits, how much should it charge per
month for a lobstering permit? (Hint: With a license fee of Fthousand
dollars per month, the marginal cost of operating a boat for a month
35.2 (2) Suppose that a honey farm is located next to an apple orchard
and each acts as a competitive firm. Let the amount of apples produced
be measured by Aand the amount of honey produced be measured by H.
The cost functions of the two firms are cH(H)=H2/100 and cA(A)=
A2/100 −H. The price of honey is $2 and the price of apples is $3.
(a) If the firms each operate independently, the equilibrium amount of
NAME 429
(b) Suppose that the honey and apple firms merged. What would be
(c) What is the socially efficient output of honey? 150. If the firms
stayed separate, how much would honey production have to be subsidized
35.3 (2) In El Carburetor, California, population 1,001, there is not
much to do except to drive your car around town. Everybody in town
is just like everybody else. While everybody likes to drive, everybody
complains about the congestion, noise, and pollution caused by traffic. A
typical resident’s utility function is U(m, d, h)=m+16d−d2−6h/1,000,
where mis the resident’s daily consumption of Big Macs, dis the number
of hours per day that he, himself, drives, and his the total amount of
driving (measured in person-hours per day) done by all other residents
of El Carburetor. The price of Big Macs is $1 each. Every person in El
Carburetor has an income of $40 per day. To keep calculations simple,
suppose it costs nothing to drive a car.
(a) If an individual believes that the amount of driving he does won’t af-
fect the amount that others drive, how many hours per day will he choose
to drive? 8. (Hint: What value of dmaximizes U(m, d, h)?)
(b) If everybody chooses his best d, then what is the total amount hof
(d) If everybody drives 6 hours a day, what will be the utility level of a
(e) Suppose that the residents decided to pass a law restricting the total
number of hours that anyone is allowed to drive. How much driving
should everyone be allowed if the objective is to maximize the utility of
the typical resident? (Hint: Rewrite the utility function, substituting
430 EXTERNALITIES (Ch. 35)
(f) The same objective could be achieved with a tax on driving. How
much would the tax have to be per hour of driving? (Hint: This price
would have to equal an individual’s marginal rate of substitution between
35.4 (3) Tom and Jerry are roommates. They spend a total of 80 hours
a week together in their room. Tom likes loud music, even when he sleeps.
His utility function is UT(CT,M)=CT+M,whereCTis the number
of cookies he eats per week and Mis the number of hours of loud music
per week that is played while he is in their room. Jerry hates all kinds
of music. His utility function is U(CJ,M)=CJ−M2/12. Every week,
Tom and Jerry each get two dozen chocolate chip cookies sent from home.
They have no other source of cookies. We can describe this situation with
a box that looks like an Edgeworth box. The box has cookies on the
horizontal axis and hours of music on the vertical axis. Since cookies are
private goods, the number of cookies that Tom consumes per week plus
the number that Jerry consumes per week must equal 48. But music in
their room is a public good. Each must consume the same number of
hours of music, whether he likes it or not. In the box, let the height of a
point represent the total number of hours of music played in their room
per week. Let the distance of the point from the left side of the box be
“cookies for Tom” and the distance of the point from the right side of the
box be “cookies for Jerry.”
0122436
48
20
40
60
Cookies
Music
80
Blue Line
Red Line
Blue Shading
a
b
Blue Line
Red Line
Tom
Jerry
(a) Suppose the dorm’s policy is that you must have your roommate’s
permission to play music. The initial endowment in this case denotes the
situation if Tom and Jerry make no deals. There would be no music, and
each person would consume 2 dozen cookies a week. Mark this initial
endowment on the box above with the label A. Use red ink to sketch
the indifference curve for Tom that passes through this point, and use
NAME 431
blue ink to sketch the indifference curve for Jerry that passes through
this point. [Hint: When you draw Jerry’s indifference curve, remember
two things: (1) He hates music, so he prefers lower points on the graph
(b) Suppose, alternatively, that the dorm’s policy is “rock-n-roll is good
for the soul.” You don’t need your roommate’s permission to play music.
Then the initial endowment is one in which Tom plays music for all of
the 80 hours per week that they are in the room together and where each
consumes 2 dozen cookies per week. Mark this endowment point in the
Calculus 35.5 (0) A clothing store and a jewelry store are located side by side
in a small shopping mall. The number of customers who come to the
shopping mall intending to shop at either store depends on the amount
of money that the store spends on advertising per day. Each store also
attracts some customers who came to shop at the neighboring store. If
the clothing store spends $xCper day on advertising, and the jeweller
spends $xJon advertising per day, then the total profits per day of the
clothing store are ΠC(xC,x
J)=(60+xJ)xC−2x2
C, and the total profits
per day of the jewelry store are ΠJ(xC,x
J) = (105 + xC)xJ−2x2
J.(In
each case, these are profits net of all costs, including advertising.)
(a) If each store believes that the other store’s amount of advertising
is independent of its own advertising expenditure, then we can find the
equilibrium amount of advertising for each store by solving two equations
in two unknowns. One of these equations says that the derivative of the
clothing store’s profits with respect to its own advertising is zero. The
other equation requires that the derivative of the jeweller’s profits with
respect to its own advertising is zero. These two equations are written as
432 EXTERNALITIES (Ch. 35)
(b) The extra profit that the jeweller would get from an extra dollar’s
worth of advertising by the clothing store is approximately equal to the
derivative of the jeweller’s profits with respect to the clothing store’s ad-
vertising expenditure. When the two stores are doing the equilibrium
amount of advertising that you calculated above, a dollar’s worth of ad-
vertising by the clothing store would give the jeweller an extra profit of
(c) Suppose that the owner of the clothing store knows the profit functions
of both stores. She reasons to herself as follows. Suppose that I can
decide how much advertising I will do before the jeweller decides what
he is going to do. When I tell him what I am doing, he will have to
adjust his behavior accordingly. I can calculate his reaction function to
my choice of xC, by setting the derivative of his profits with respect to his
own advertising equal to zero and solving for his amount of advertising
as a function of my own advertising. When I do this, I find that xJ=
(d) Suppose that the clothing store and the jewelry store have the same
profit functions as before but are owned by a single firm that chooses
the amounts of advertising so as to maximize the sum of the two stores’
profits. The single firm would choose xC=$37.50 and xJ=
$45. Without calculating actual profits, can you determine whether
total profits will be higher, lower, or the same as total profits would be
35.6 (2) The cottagers on the shores of Lake Invidious are an unsavory
bunch. There are 100 of them, and they live in a circle around the lake.
Each cottager has two neighbors, one on his right and one on his left.
There is only one commodity, and they all consume it on their front
lawns in full view of their two neighbors. Each cottager likes to consume
the commodity but is very envious of consumption by the neighbor on
his left. Curiously, nobody cares what the neighbor on his right is doing.
In fact every consumer has a utility function U(c, l)=c−l2,wherecis
his own consumption and lis consumption by his neighbor on the left.
